proposition 45.46 The first law

open in the book · parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:1743 · p. 1304

Rests on

depends_on proposition 45.43 Static acceleration and surface gravity ¶
- depends_on theorem 45.30 Kerr solution; imported ¶
  - depends_on theorem 45.1 Schwarzschild solution ¶
    - depends_on equation 13.306 eq:mfd-riemann ¶
    - depends_on postulate 43.1 Geodesic motion ¶
    - depends_on theorem 13.150 Levi-Civita connection and contorsion ¶
      - depends_on definition 13.145 Affine connection ¶
      - depends_on definition 13.117 Metric tensor of signature $(p,q)$ ¶
      - depends_on definition 13.149 Metric compatibility ¶
      - depends_on definition 13.147 Torsion ¶
      - proves proof ch:11-manifolds-tensors-curvature@proof-41 ¶
    - proves proof ch:04-schwarzschild-black-holes@proof-1 ¶
- depends_on theorem 45.1 Schwarzschild solution ¶ ↺
- proves proof ch:04-schwarzschild-black-holes@proof-20 ¶
depends_on proposition 45.45 Irreducible mass and the Christodoulou–Smarr mass formula ¶
- depends_on lemma 45.44 Horizon area ¶
  - depends_on proposition 45.31 Horizons and the extremal bound ¶
    - depends_on theorem 45.30 Kerr solution; imported ¶ ↺
    - proves proof ch:04-schwarzschild-black-holes@proof-15 ¶
  - proves proof ch:04-schwarzschild-black-holes@proof-21 ¶
- depends_on proposition 45.31 Horizons and the extremal bound ¶ ↺
- proves proof ch:04-schwarzschild-black-holes@proof-22 ¶
proves proof ch:04-schwarzschild-black-holes@proof-23 ¶

Supports

depends_on theorem 45.49 Hawking temperature; imported ¶
- depends_on proposition 45.51 Evaporation: scale and timescale ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`cites`	→	Mass Formula for Kerr Black Holes	derived	parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:1752
`depends_on`	→	Static acceleration and surface gravity	declared	parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:1757
`depends_on`	→	Irreducible mass and the Christodoulou–Smarr mass formula	declared	parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:1757
`depends_on`	←	Hawking temperature; imported	declared	parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:1883
`proves`	←	ch:04-schwarzschild-black-holes@proof-23	declared	parts/05-general-relativity-cosmology/04-schwarzschild-black-holes.tex:1760